2.

5 Continuity

Recall that in previous section we talked about how we can evaluate the limit of a polynomial at a number by simply
directly substituting that number in. This is actually a special case of continuous functions. Let us have a look at
another example.

Ex 1. Evaluate the limit.

lim sin x
x→π

1 y
f (x) = sin x
0.5

x
−2π − 7π
4
− 3π
2
− 5π
4 −π − 3π
4
−π
2
−π
4
π
4
π
2
3π
4
π 5π
4
3π
2
7π
4 2π
−0.5

−1

Definition. A function is said to be continuous at a if

lim f (x) = f (a)

x→a

Notice that the above definition implies the following statements as a consequence:

1. f (a) is defined.
2. lim f (x) exists.
x→a

3. lim f (x) = f (a).

x→a

Note. We call a function discontinuous at a if it is not continuous at x = a. Can you think of what sort of
functions are discontinuous?
Types of Discontinuities
y y y
2 2 2

1 1 1
x x x
−2 −1 1 2 −2 −1 1 2 −2 −1 1 2
−1 −1 −1

Definition. A function is said to be continuous from the right at a if

lim f (x) = f (a)

x→a+

and f is said to be continuous from the left at a if

lim f (x) = f (a)

x→a−

A function is said to be continuous on an interval [a, b] if it continuous at every number in that interval.

2
x + 1
 x<1
Ex 2. Is the following function continuous, continuous from left/right, or neither at x = 1? f (x) = 3 x=1

2+x x>1


Ex 3. Looking at the graph of f, state x-values at which f is discontinuous. For each x-value, determine whether f
is continuous from the right, or from the left, or neither.
Theorem. If f , g are continuous at a and say c is a constant, then the following functions are also continuous at a:

• f +g • f −g • cf
f
• f.g • g if g(a) 6= 0

Theorem. Using the theorem above we conclude,

(a)Polynomials are continuous everywhere.
(b)A Rational function is continuous wherever it is defined.

Ex 4. Can you give examples of functions that are continuous on their domain?

x3 + 2x2 − 1
Ex 5. Find lim
x→−2 5 − 3x

ln(x) + tan−1 x
Ex 6. Find the interval(s) at which f (x) = is continuous.
x−1
Theorem.If f is continuous at b and lim g(x) = b, then lim f (g(x)) = f (b). In other words,
x→a x→a

lim f (g(x)) = f ( lim g(x))

x→a x→a

5x3 + x

Ex 7. Use the above theorem to evaluate lim cos .
x→0 3x + 1

2
−2x
Ex 8. Evaluate lim e3x .
x→2

Theorem. If g is continuous at a and f is continuous at b = g(a) then the composition (f ◦ g)(x) = f (g(x)) is
continuous at a as well.

Ex 9. Find the interval at which f (x) = ln(1 − x2 ) is continuous.

The Intermediate Value Theorem.Suppose f is continuous on [a, b] and N is an arbitrary number between f (a)
and f (b), then there has to be a number c ∈ [a, b] such that f (c)=N.

Ex 10. Consider the equation 4x3 − 6x2 + 3x − 2 = 0. Apply the IVT to deduce that the equation has at least one
solution on the interval [1, 2].

Ex 11. Could you use IVT to prove that at some point in your life your height was 4 ft? How so, discuss.
Exercises

Here are some extra problems. I will put the answers on blackboard under Lecture Note-Solutions tab.

1. Find a value of the constant c so that the following function is continuous at x = 1.

(
cx2 + 1 x < 1
f (x) =
2x − c x ≥ 1

3x
sin( πx
2 )e +x
2. (a)Consider the function f (x) = x2 +2 . Where is this function continuous?

(b)Compute lim f (x). Use part-a to do this.

x→1

3. Consider the equation x4 + x − 3 = 0. Apply the IVT to deduce that the equation has at least one solution on
the interval [1, 2].

√
4. Using the limit laws, show that f (t) = 4 6t2 + 1 is continuous at t = 2 step by step.